Compact difference approximation with consistent boundary condition

For simulating multi-scale complex flow fields it should be noted that all the physical quantities we are interested in must be simulated well. With limitation of the computer resources it is preferred to use high order accurate difference schemes. Because of their high accuracy and small stencil of grid points computational fluid dynamics (CFD) workers pay more attention to compact schemes recently. For simulating the complex flow fields the treatment of boundary conditions at the far field boundary points and near far field boundary points is very important. According to authors' experience and published results some aspects of boundary condition treatment for far field boundary are presented, and the emphasis is on treatment of boundary conditions for the upwind compact schemes. The consistent treatment of boundary conditions at the near boundary points is also discussed. At the end of the paper are given some numerical examples. The computed results with presented method are satisfactory.

Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Upwind compact method and direct numerical-simulation of driven flow in a square cavity

A high-order accurate finite-difference scheme, the upwind compact method, is proposed. The 2-D unsteady incompressible Navier-Stokes equations are solved in primitive variables. The nonlinear convection terms in the governing equations are approximated by using upwind biased compact difference, and other spatial derivative terms are discretized by using the fourth-order compact difference. The upwind compact method is used to solve the driven flow in a square cavity. Solutions are obtained for Reynolds numbers as high as 10000. When Re less than or equal to 5000, the results agree well with those in literature. When Re = 7500 and Re = 10000, there is no convergence to a steady laminar solution, and the flow becomes unsteady and periodic.

Upwind compact scheme with dispersion control for aerodynamic equations

A compact upwind scheme with dispersion control is developed using a dissipation analogy of the dispersion term. The term is important in reducing the unphysical fluctuations in numerical solutions. The scheme depends on three free parameters that may be used to regulate the size of dissipation as well as the size and direction of dispersion. A coefficient to coordinate the dispersion is given. The scheme has high accuracy, the method is simple, and the amount of computation is small. It also has a good capability of capturing shock waves. Numerical experiments are carried out with two-dimensional shock wave reflections and the results are very satisfactory.